Rotation of Ellipse: What would be the best way to simplify this equation to its standard form 7(sq root 3x - y/2)^2 -6sq root 3(sq root 3x-y/2)(x+sq root 3y/2)+13(x+sq root 3y/2)^2-16=0

Answer 1

Note that the /2 are fractions that work for both terms.

The eigenvalues for

7 -3 √3

-3√3 13

are 4 and 16, with eigenvectors √3/2 1/2 and -1/2 √3/2

Thus, as the axes in the polynomial are oriented √3/2 -1/2 and 1/2 √3/2, we see, examining these axes and eigenvectors and using the eigenvalues, that the expanded term becomes

4x^2 + 16y^2 = 16, or

x^2/4 + y^2 = 1